Let $S$ be a wedge of the cylinder with height $4$ and radius $2$ such that $\dfrac{\pi}{3} < \theta < \dfrac{2\pi}{3}$, whose axis is parallel to the $z$ -axis and whose lower base is centered at the origin. What is the triple integral of the scalar field $f(x, y, z) = xy + z$ over $S$ in cylindrical coordinates? Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} r^3\sin(2\theta) + rz \, d\theta \, dr \, dz$ (Choice B) B $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} r^3\sin(\theta)\cos(\theta) + rz \, d\theta \, dr \, dz$ (Choice C) C $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} r^3(\sin(\theta) + z) \, d\theta \, dr \, dz$ (Choice D) D $ \int_0^2 \int_0^4 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} r^2\sin(\theta)\cos(\theta) + z \, d\theta \, dr \, dz$
Solution: The only bound is $0 < \theta < 2\pi$. Here is the change of variables for cylindrical coordinates. $\begin{aligned} x &= r \cos(\theta) \\ \\ y &= r \sin(\theta) \\ \\ z &= z \end{aligned}$ We want to represent the cylinder $S$ with bounds in cylindrical coordinates. Here, the region $S$ ranges from a height of $z = 0$ to $z = 4$ and radius of $r = 0$ to $r = 2$. Theta goes from $\dfrac{\pi}{3}$ to $\dfrac{2\pi}{3}$ so that we only integrate over the specified wedge. $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \cdots \, d\theta \, dr \, dz$ We can now put $f(x, y, z)$ in the integrand, but we need to substitute $x$, $y$, and $z$ for their definitions in cylindrical coordinates. $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} (r^2\sin(\theta)\cos(\theta) + z) \cdots \, d\theta \, dr \, dz$ The final step is finding the Jacobian of cylindrical coordinates, which we'll need to multiply in to get the final integral. $J(r, \theta, z) = r$ [Derivation] The integral in cylindrical coordinates: $ \int_0^4 \int_0^2 \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} r^3\sin(\theta)\cos(\theta) + rz \, d\theta \, dr \, dz$